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Chapter 2: Problem 2

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=2, K=50\)

Short Answer

Expert verified
The recursion is \(N_{t+1} = \frac{100 N_t}{50 + N_t}\). The graph approaches 1 as \(N_t\) nears 50.

Step by step solution

01

Understand the Beverton-Holt Model

The Beverton-Holt model describes population dynamics with a growth parameter \(R\) and carrying capacity \(K\). The recursion relation for population \(N_t\) is defined as \(N_{t+1} = \frac{R N_t}{1 + \left(\frac{R-1}{K}\right) N_t}\). Here, \(R=2\) and \(K=50\) are given.
02

Substitute Given Values into the Recursion Relation

Substitute \(R=2\) and \(K=50\) into the Beverton-Holt model equation: \[N_{t+1} = \frac{2 N_t}{1 + \left(\frac{2-1}{50}\right) N_t} = \frac{2 N_t}{1 + \frac{1}{50} N_t}\].
03

Simplify the Equation

Simplify the equation \(N_{t+1} = \frac{2 N_t}{1 + \frac{1}{50} N_t}\) by writing it as: \[N_{t+1} = \frac{2 N_t}{1 + \frac{N_t}{50}} = \frac{2 N_t}{\frac{50 + N_t}{50}} = \frac{100 N_t}{50 + N_t}\].
04

Graph \(N_t/N_{t+1}\) versus \(N_t\)

The graph of the function \( N_t / N_{t+1} \) as a function of \( N_t \) can be expressed as \( \frac{N_t}{N_{t+1}} = \frac{50 + N_t}{100} \). Plotting this, you will see that as \(N_t\) increases, \(N_t/N_{t+1}\) approaches 1.
05

Analyze the Graph

The graph shows that for small values of \(N_t\), the ratio \(N_t/N_{t+1}\) is less than 1, indicating growth. As \(N_t\) approaches \(K=50\), the ratio approaches 1, reflecting that \(N_t\) stabilizes at the carrying capacity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Dynamics in the Beverton-Holt Model
Population dynamics studies how and why the number of individuals in a population changes over time. The Beverton-Holt model is a mathematical representation used to describe these changes specifically for populations with capped growth due to limited resources.

This model considers that populations grow until they reach a "carrying capacity". Initially, populations grow quickly due to abundant resources but slow down as resources become scarce. This is because the environment can only support a limited number of individuals.
  • By representing the change from generation to generation, the model gives a clear picture of how populations stabilize over time.
  • It helps in understanding factors like resource availability and individual survival rates which affect population growth.
Thus, studying population dynamics through models like Beverton-Holt provides crucial insights into conserving resources and planning sustainable development.
Understanding the Growth Parameter
The growth parameter, denoted as \( R \) in the Beverton-Holt model, reflects how quickly a population can grow when resources are not limiting. In simple terms, it tells us how effective the population is at reproducing.

In our specific example, \( R = 2 \) which means that under ideal conditions, the population can potentially double each generation. This is a measure of biological potential, assuming no constraints on resources.
  • When \( R > 1 \), it indicates potential for growth since the population size can increase each generation.
  • If \( R = 1 \), the population is stable鈥攏either growing nor shrinking under ideal conditions.
  • If \( R < 1 \), the population would shrink, indicating poor reproductive success.
By understanding the growth parameter, we can predict how quickly a population might expand if environmental conditions were perfect.
Exploring the Carrying Capacity
Carrying capacity, denoted as \( K \), represents the maximum population size that an environment can sustain indefinitely. Beyond this size, resources become too scarce to support additional individuals.

In the model, once a population nears its carrying capacity, growth slows down largely due to competition for resources.
  • In our scenario, \( K = 50 \), indicating the environment in question can house up to 50 individuals efficiently.
  • As the population nears this limit, competition for resources like food and space increases, slowing further growth.
  • When the population consistently hovers around \( K \), it means the population has reached an equilibrium with what the environment can support.
Understanding \( K \) is crucial for managing wildlife conservation and setting sustainable goals.
Analyzing the Recursion Relation
The recursion relation in the Beverton-Holt model is an equation that describes how the population changes from one generation to the next. It is the core mathematical formula that encapsulates both the growth parameter and the carrying capacity.

The recursion relation for the Beverton-Holt model is \( N_{t+1} = \frac{RN_t}{1 + (\frac{R-1}{K}) N_t} \). This tells us how the population \( N \) at time \( t+1 \) depends on the population at time \( t \).
  • The equation allows us to calculate the future population size based on current size, growth potential, and environmental constraints.
  • It shows how population growth is initially rapid but slows as resources are depleted near \( K \).
  • By plotting these values, we can visualize population trends and predict future changes.
Recursion relations are vital for simulation models, helping ecologists plan and adjust conservation efforts efficiently.

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Most popular questions from this chapter

Formal Definition of Limits: In Problems \(65-70\), use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{t \rightarrow \infty} \frac{1}{n}=0 $$

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=2, K=150\)

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t\). \(r=3.1, x_{0}=0\)

Compute \(N_{t}\) and \(N_{t} / N_{t-1}\) for \(t=2,3,4, \ldots, 20\) when $$ N_{t+1}=N_{t}+N_{t-1} $$ with \(N_{0}=1\) and \(N_{1}=1\).

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n+1}{n^{2}-1}\right) $$
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